For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation}

consisting of all products of all non-zero linear forms $a_1x_1+\ldots +a_kx_k$, at most $\ell$ of whose coefficients are non-zero.

Now let \begin{equation} \alpha_j = \sum_{i=0}^{\ell-1}\binom{j-1}{i}, \end{equation}

and

\begin{equation} D_{k,\ell}= \det \begin{pmatrix} x_1^{\alpha_1} & \cdots & x_k^{\alpha_1} \\ \vdots & \vdots & \vdots \\ x_1^{\alpha_k} & \cdots & x_k^{\alpha_k} \end{pmatrix} = \sum_{\sigma \in S_k} x_{\sigma(1)}^{\alpha_1} x_{\sigma(2)}^{\alpha_2}\cdots x_{\sigma(k)}^{\alpha_k}, \end{equation}

where $S_k$ is the symmetric group.

(Note that $\alpha_{1}=1$, and we assume as usual that $\binom{a}{b}=0$ if $a<b$.)

Question: It is a classical fact that $D_{k,\ell}=P_{k,\ell}$ when $\ell =2, k$. What I would like to know is if it is true that, for arbitrary $2\leq \ell \leq k$, that $P_{k,\ell}$ is just $D_{k,\ell}$ plus (possibly) some additional terms.

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation}

consisting of all products of all non-zero linear forms $a_1x_1+\ldots +a_kx_k$, at most $\ell$ of whose coefficients are non-zero.

Now let \begin{equation} \alpha_j = \sum_{i=0}^{\ell-1}\binom{j-1}{i}, \end{equation}

and

\begin{equation} D_{k,\ell}= \det \begin{pmatrix} x_1^{\alpha_1} & \cdots & x_k^{\alpha_1} \\ \vdots & \vdots & \vdots \\ x_1^{\alpha_k} & \cdots & x_k^{\alpha_k} \end{pmatrix} = \sum_{\sigma \in S_k} x_{\sigma(1)}^{\alpha_1} x_{\sigma(2)}^{\alpha_2}\cdots x_{\sigma(k)}^{\alpha_k}, \end{equation}

where $S_k$ is the symmetric group.

(Note that $\alpha_{1}=1$, and we assume as usual that $\binom{a}{b}=0$ if $a<b$.)

Question: It is a classical fact that $D_{k,\ell}=P_{k,\ell}$ when $\ell =2, k$. Is it true that, for arbitrary $2\leq \ell \leq k$, $D_{k,\ell}$ is $P_{k,\ell}$ plus (possibly) some additional terms?

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation}

consisting of all products of all non-zero linear forms $a_1x_1+\ldots +a_kx_k$, at most $\ell$ of whose coefficients are non-zero.

Letting \begin{equation} \alpha_j = \sum_{i=0}^{\ell-1}\binom{j-1}{i}, \end{equation} it is not hard to confirm that $P_{k,\ell}$ can be expressed as the following determinant:

\begin{equation} P_{k,\ell}= \det \begin{pmatrix} x_1^{\alpha_1} & \cdots & x_k^{\alpha_1} \\ \vdots & \vdots & \vdots \\ x_1^{\alpha_k} & \cdots & x_k^{\alpha_k} \end{pmatrix} = \sum_{\sigma \in S_k} x_{\sigma(1)}^{\alpha_1} x_{\sigma(2)}^{\alpha_2}\cdots x_{\sigma(k)}^{\alpha_k}, \end{equation}

$S_k$ the symmetric group.

(Note that $\alpha_{1}=1$, and we assume as usual that $\binom{a}{b}=0$ if $a<b$.)

Question: I am sure the formula must be classically known (e.g., it is a Dickson polynomial when $\ell=k$). I would be grateful for a reference.

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation}

consisting of all products of all non-zero linear forms $a_1x_1+\ldots +a_kx_k$, at most $\ell$ of whose coefficients are non-zero.

Now let \begin{equation} \alpha_j = \sum_{i=0}^{\ell-1}\binom{j-1}{i}, \end{equation}

and

\begin{equation} D_{k,\ell}= \det \begin{pmatrix} x_1^{\alpha_1} & \cdots & x_k^{\alpha_1} \\ \vdots & \vdots & \vdots \\ x_1^{\alpha_k} & \cdots & x_k^{\alpha_k} \end{pmatrix} = \sum_{\sigma \in S_k} x_{\sigma(1)}^{\alpha_1} x_{\sigma(2)}^{\alpha_2}\cdots x_{\sigma(k)}^{\alpha_k}, \end{equation}

where $S_k$ is the symmetric group.

(Note that $\alpha_{1}=1$, and we assume as usual that $\binom{a}{b}=0$ if $a<b$.)

Question: It is a classical fact that $D_{k,\ell}=P_{k,\ell}$ when $\ell =2, k$. What I would like to know is if it is true that, for arbitrary $2\leq \ell \leq k$, that $P_{k,\ell}$ is just $D_{k,\ell}$ plus (possibly) some additional terms.